cqplot: Chi Square Quantile-Quantile plots

cqplotR Documentation

Chi Square Quantile-Quantile plots

Description

A chi square quantile-quantile plots show the relationship between data-based values which should be distributed as \chi^2 and corresponding quantiles from the \chi^2 distribution. In multivariate analyses, this is often used both to assess multivariate normality and check for or identify outliers.

For a data frame of numeric variables or a matrix supplied as the argument x, it uses the Mahalanobis squared distances (D^2) of observations \mathbf{x} from the centroid \bar{\mathbf{x}} taking the sample covariance matrix \mathbf{S} into account,

D^2 = (\mathbf{x} - \bar{\mathbf{x}})^\prime \; \mathbf{S}^{-1} \; (\mathbf{x} - \bar{\mathbf{x}}) \; .

The method for "mlm" objects fit using lm for a multivariate response applies this to the residuals from the model.

Usage

cqplot(x, ...)

## S3 method for class 'mlm'
cqplot(x, ...)

## Default S3 method:
cqplot(
  x,
  method = c("classical", "mcd", "mve"),
  detrend = FALSE,
  pch = 19,
  col = palette()[1],
  cex = par("cex"),
  ref.col = "red",
  ref.lwd = 2,
  conf = 0.95,
  env.col = "gray",
  env.lwd = 2,
  env.lty = 1,
  env.fill = TRUE,
  fill.alpha = 0.2,
  fill.color = trans.colors(ref.col, fill.alpha),
  labels = if (!is.null(rownames(x))) rownames(x) else 1:nrow(x),
  id.n,
  id.method = "r",
  id.cex = 1,
  id.col = palette()[1],
  xlab,
  ylab,
  main,
  what = deparse(substitute(x)),
  ylim,
  ...
)

Arguments

x

either a numeric data frame or matrix for the default method, or an object of class "mlm" representing a multivariate linear model. In the latter case, residuals from the model are plotted.

...

Other arguments passed to methods

method

estimation method used for center and covariance, one of: "classical" (product-moment), "mcd" (minimum covariance determinant), or "mve" (minimum volume ellipsoid).

detrend

logical; if FALSE, the plot shows values of D^2 vs. \chi^2. if TRUE, the ordinate shows values of D^2 - \chi^2

pch

plot symbol for points. Can be a vector of length equal to the number of rows in x.

col

color for points. Can be a vector of length equal to the number of rows in x. The default is the first entry in the current color palette (see palette and par).

cex

character symbol size for points. Can be a vector of length equal to the number of rows in x.

ref.col

Color for the reference line

ref.lwd

Line width for the reference line

conf

confidence coverage for the approximate confidence envelope

env.col

line color for the boundary of the confidence envelope

env.lwd

line width for the confidence envelope

env.lty

line type for the confidence envelope

env.fill

logical; should the confidence envelope be filled?

fill.alpha

transparency value for fill.color

fill.color

color used to fill the confidence envelope

labels

vector of text strings to be used to identify points, defaults to rownames(x) or observation numbers if rownames(x) is NULL

id.n

number of points labeled. If id.n=0, the default, no point identification occurs.

id.method

point identification method. The default id.method="r" will identify the id.n points with the largest value of abs(y), i.e., the largest Mahalanobis DSQ. See showLabels for other options.

id.cex

size of text for point labels

id.col

color for point labels

xlab

label for horizontal (theoretical quantiles) axis

ylab

label for vertical (empirical quantiles) axis

main

plot title

what

the name of the object plotted; used in the construction of main when that is not specified.

ylim

limits for vertical axis. If not specified, the range of the confidence envelope is used.

Details

cqplot is a more general version of similar functions in other packages that produce chi square QQ plots. It allows for classical Mahalanobis squared distances as well as robust estimates based on the MVE and MCD; it provides an approximate confidence (concentration) envelope around the line of unit slope, a detrended version, where the reference line is horizontal, the ability to identify or label unusual points, and other graphical features.

Cases with any missing values are excluded from the calculation and graph with a warning.

Confidence envelope

In the typical use of QQ plots, it essential to have something in the nature of a confidence band around the points to be able to appreciate whether, and to what degree the observed data points differ from the reference distribution. For cqplot, this helps to assess whether the data are reasonably distributed as multivariate normal and also to flag potential outliers.

The calculation of the confidence envelope here follows that used in the SAS program, http://www.datavis.ca/sasmac/cqplot.html which comes from Chambers et al. (1983), Section 6.8.

The essential formula computes the standard errors as:

\text{se} ( D^2_{(i)} ) = \frac{\hat{b}} {d ( q_i)} \times \sqrt{ p_i (1-p_i) / n }

where D^2_{(i)} is the i-th ordered value of D^2, \hat{b} is an estimate of the slope of the reference line obtained from the ratio of the interquartile range of the D^2 values to that of the \chi^2_p distribution and d(q_i) is the density of the chi square distribution at the quantile q_i.

The pointwise confidence envelope of coverage conf = 1-\alpha is then calculated as D^2_{(i)} \pm z_{1-\alpha/2} \text{se} ( D^2_{(i)} )

Note that this confidence envelope applies only to the D^2 computed using the classical estimates of location (\bar{\mathbf{x}}) and scatter (\mathbf{S}). The qqPlot function provides for simulated envelopes, but only for a univariate measure. Oldford (2016) provides a general theory and methods for QQ plots.

Value

Returns invisibly a data.frame containing squared Mahalanobis distances (DSQ), their quantiles and p-values corresponding to the rows of x or the residuals of the model for the identified points, else NULL if no points are identified.

Author(s)

Michael Friendly

References

J. Chambers, W. S. Cleveland, B. Kleiner, P. A. Tukey (1983). Graphical methods for data analysis, Wadsworth.

R. W. Oldford (2016), "Self calibrating quantile-quantile plots", The American Statistician, 70, 74-90.

See Also

Mahalanobis for calculation of Mahalanobis squared distance;

qqplot; qqPlot can give a similar result for Mahalanobis squared distances of data or residuals; qqtest has many features for all types of QQ plots.

Examples



cqplot(iris[, 1:4])

iris.mod <- lm(as.matrix(iris[,1:4]) ~ Species, data=iris)
out <- cqplot(iris.mod, id.n=3)
# show return value
out

# compare with car::qqPlot
car::qqPlot(Mahalanobis(iris[, 1:4]), dist="chisq", df=4)


# Adopted data
Adopted.mod <- lm(cbind(Age2IQ, Age4IQ, Age8IQ, Age13IQ) ~ AMED + BMIQ, 
                  data=Adopted)
cqplot(Adopted.mod, id.n=3)
cqplot(Adopted.mod, id.n=3, method="mve")


# Sake data
Sake.mod <- lm(cbind(taste, smell) ~ ., data=Sake)
cqplot(Sake.mod)
cqplot(Sake.mod, method="mve", id.n=2)

# SocialCog data -- one extreme outlier
data(SocialCog)
SC.mlm <-  lm(cbind(MgeEmotions,ToM, ExtBias, PersBias) ~ Dx,
               data=SocialCog)
cqplot(SC.mlm, id.n=1)

# data frame example: stackloss data
data(stackloss)
cqplot(stackloss[, 1:3], id.n=4)                # very strange
cqplot(stackloss[, 1:3], id.n=4, detrend=TRUE)
cqplot(stackloss[, 1:3], id.n=4, method="mve")
cqplot(stackloss[, 1:3], id.n=4, method="mcd")




heplots documentation built on April 17, 2025, 1:08 a.m.